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3.2 Addition & Subtraction of Decimals

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ACCUPLACER Arithmetic Exam

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1 Chapter 1: Whole Number Operations

1.1 Basic Whole Number Arithmetic 1.2 Order of Operations & Grouping Symbols 1.3 Number Properties, Factors & Prime Factorization 1.4 Whole Number Estimation, Rounding & Applied Word Problems

2 Chapter 2: Fraction Operations

2.1 Fraction Fundamentals & Simplifying 2.2 Addition & Subtraction of Fractions and Mixed Numbers 2.3 Multiplication & Division of Fractions and Mixed Numbers 2.4 Real-World Fraction Applications & Word Problems

3 Chapter 3: Decimal Operations

3.1 Decimal Place Value, Rounding & Comparing Decimals 3.2 Addition & Subtraction of Decimals 3.3 Multiplication & Division of Decimals 3.4 Decimal Word Problems & Financial Calculations

4 Chapter 4: Percent

4.1 Percent Concepts & Conversions 4.2 Percent of a Quantity & Percent Equations 4.3 Percent Increase, Decrease & Percent Change 4.4 Practical Percent Applications

5 Chapter 5: Number Comparisons & Equivalents

5.1 Ordering Mixed Numerical Forms 5.2 Equivalent Numerical Representations & Conversion Strategies 5.3 Ratios, Unit Rates & Proportional Reasoning

3.2 Addition & Subtraction of Decimals

Key Takeaways Explain This Topic AI

The fundamental rule for adding and subtracting decimals is lining up the decimal points vertically so that identical place values align in columns.

Annexing trailing zeros to numbers with fewer decimal places prevents column misalignment errors and simplifies subtraction regrouping.

Regrouping (borrowing) across zeros in decimal subtraction follows standard base-10 rules across the decimal point.

When finding perimeters or solving multi-step decimal problems, ensure all measurements use identical units before performing addition or subtraction.

The decimal point in the final sum or difference drops directly straight down into the aligned vertical column of the input decimal points.

Last updated: August 2026

3.2 Addition & Subtraction of Decimals

Adding and subtracting decimal numbers requires the exact same column-by-column regrouping rules as whole number arithmetic, with one critical prerequisite: you must line up the decimal points vertically . Aligning decimal points ensures that digits of equal place value—tenths with tenths, hundredths with hundredths, ones with ones—are added or subtracted together.

The Core Rule: Vertical Alignment of Decimal Points

When adding or subtracting whole numbers, numbers are naturally right-justified because the ones place is always the rightmost digit. With decimal numbers, right-justification fails because numbers can have different numbers of digits after the decimal point.

Alignment Comparison

Consider adding 14.2 , 3.85 , and 0.174 :

  • INCORRECT (Right-Justified): 14.2 3.85 + 0.174 Wrong Column Sums \begin{array}{r}
  • 14.2 \\
  • 3.85 \\
  • + 0.174 \\
  • \hline
  • \text{Wrong Column Sums}
  • \end{array} 14.2 3.85 + 0.174 Wrong Column Sums ​ ​

Error: Adds 2 (tenths) to 5 (hundredths) to 4 (thousandths).

CORRECT (Decimal Points Aligned Vertically): 14.200 ( annexed zeros ) 3.850 ( annexed zero ) + 0.174 ( aligned ) 18.224 ( decimal drops straight down ) \begin{array}{rl}

  • 14.200 & (\text{annexed zeros}) \\
  • 3.850 & (\text{annexed zero}) \\
  • + 0.174 & (\text{aligned}) \\
  • \hline
  • \mathbf{18.224} & (\text{decimal drops straight down})

\end{array} 14.200 3.850 + 0.174 18.224 ​ ( annexed zeros ) ( annexed zero ) ( aligned ) ( decimal drops straight down ) ​ ​

Annexing Trailing Zeros for Subtraction

Annexing zeros means adding zeros to the right of the last digit in a decimal fraction. Because 0.7 = 0.70 = 0.700 0.7 = 0.70 = 0.700 0.7 = 0.70 = 0.700 , annexing zeros never alters the value of a number, but it is indispensable when subtracting a decimal with more decimal places from one with fewer decimal places.

Step-by-Step Subtraction with Annexed Zeros

Subtract 9.175 from 25.4 .

Step 1: Write the top number (25.4) and annex two zeros to match the 3 decimal places of 9.175: 25.400 − 9.175 25.400 - 9.175 25.400 − 9.175

Step 2: Line up by decimal point and set up column borrowing: 1 15 ⋅ 3 9 10 2̸ 5̸ ⋅ 4̸ 0̸ 0̸ − 0 9 ⋅ 1 7 5 1 6 ⋅ 2 2 5 \begin{array}{rl}

  • & \quad 1 \; 15 \cdot 3 \; 9 \; 10 \\
  • & \quad \not{2} \; \not{5} \cdot \not{4} \; \not{0} \; \not{0} \\
  • & \quad 0 \; 9 \cdot 1 \; 7 \; 5 \\
  • \hline
  • & \quad \mathbf{1 \; 6 \cdot 2 \; 2 \; 5}
  • \end{array} − ​ 1 15 ⋅ 3 9 10  2  5 ⋅  4  0  0 0 9 ⋅ 1 7 5 1 6 ⋅ 2 2 5 ​ ​

Step 3: Verify place values column by column:

Thousandths: Borrow from tenths to make 10 − 5 = 5 10 - 5 = 5 10 − 5 = 5 .

Hundredths: 9 − 7 = 2 9 - 7 = 2 9 − 7 = 2 .

Tenths: 3 − 1 = 2 3 - 1 = 2 3 − 1 = 2 .

Ones: Borrow 1 ten to make 15 − 9 = 6 15 - 9 = 6 15 − 9 = 6 .

Tens: 1 − 0 = 1 1 - 0 = 1 1 − 0 = 1 .

  • Result: 16.225
  • Regrouping Across Multiple Zeros

When subtracting from a number containing consecutive zeros (such as whole dollar amounts like $ 100.00 \text{\textdollar}100.00 $ 100.00 ), regrouping must cascade leftward to the first non-zero digit.

Step-by-Step Worked Example: Subtracting from Round Numbers

Calculate $100.00 - $37.84 .

  • Align decimal points vertically: 100.00 − 0 37.84 \begin{array}{r}
  • 100.00 \\
  • \phantom{0}37.84 \\
  • \hline
  • \end{array} 100.00 − 0 37.84 ​ ​

Cascade borrowing from the hundreds place:

The 1 in the hundreds place becomes 0.

The tens place becomes 9.

The ones place becomes 9.

The tenths place becomes 9.

The hundredths place becomes 10.

Perform column subtraction from right to left:

  • Hundredths: 10 − 4 = 6 10 - 4 = 6 10 − 4 = 6
  • Tenths: 9 − 8 = 1 9 - 8 = 1 9 − 8 = 1
  • Ones: 9 − 7 = 2 9 - 7 = 2 9 − 7 = 2
  • Tens: 9 − 3 = 6 9 - 3 = 6 9 − 3 = 6
  • Hundreds: 0 − 0 = 0 0 - 0 = 0 0 − 0 = 0
  • Place the decimal point straight down: Result: 62.16 \text{Result: } \mathbf{62.16} Result: 62.16
  • Practical Applications: Perimeter, Distance & Money

Decimal addition and subtraction are used in geometric measurements, inventory tracking, and personal finance calculations.

Example 1: Calculating Geometric Perimeter

Find the perimeter of an irregular quadrilateral with side lengths of 14.35 cm , 9.8 cm , 12.075 cm , and 16.4 cm .

Recall Formula: Perimeter = Side 1 + Side 2 + Side 3 + Side 4 \text{Perimeter} = \text{Side}_1 + \text{Side}_2 + \text{Side}_3 + \text{Side}_4 Perimeter = Side 1 ​ + Side 2 ​ + Side 3 ​ + Side 4 ​

  • Step 1: Line up decimal points vertically and annex zeros: 14.350 09.800 12.075 + 16.400 \begin{array}{r}
  • 14.350 \\
  • 09.800 \\
  • 12.075 \\
  • + 16.400 \\
  • \hline
  • \end{array} 14.350 09.800 12.075 + 16.400 ​ ​

Step 2: Add column by column:

  • Thousandths column: 0 + 0 + 5 + 0 = 5 0 + 0 + 5 + 0 = 5 0 + 0 + 5 + 0 = 5
  • Hundredths column: 5 + 0 + 7 + 0 = 12 5 + 0 + 7 + 0 = 12 5 + 0 + 7 + 0 = 12 (write 2, carry 1)

Tenths column: 1 (carried) + 3 + 8 + 0 + 4 = 16 1 \text{ (carried)} + 3 + 8 + 0 + 4 = 16 1 (carried) + 3 + 8 + 0 + 4 = 16 (write 6, carry 1)

Ones column: 1 (carried) + 4 + 9 + 2 + 6 = 22 1 \text{ (carried)} + 4 + 9 + 2 + 6 = 22 1 (carried) + 4 + 9 + 2 + 6 = 22 (write 2, carry 2)

Tens column: 2 (carried) + 1 + 0 + 1 + 1 = 5 2 \text{ (carried)} + 1 + 0 + 1 + 1 = 5 2 (carried) + 1 + 0 + 1 + 1 = 5

Final Answer: The perimeter is 52.625 cm .

Example 2: Financial Account Balancing

At the start of the week, Alex had a checking account balance of $524.10 . During the week, he made a cash deposit of $145.50 , paid an electric bill of $89.75 , bought groceries for $64.38 , and received a refund check of $22.15 . What is his ending account balance?

Step 1: Add all credits (deposits and refunds) to initial balance: 524.10 ( starting balance ) 145.50 ( deposit ) + 22.15 ( refund ) 691.75 ( total funds ) \begin{array}{rl}

  • 524.10 & (\text{starting balance}) \\
  • 145.50 & (\text{deposit}) \\
  • + 22.15 & (\text{refund}) \\
  • \hline
  • \mathbf{691.75} & (\text{total funds})
  • \end{array} 524.10 145.50 + 22.15 691.75 ​ ( starting balance ) ( deposit ) ( refund ) ( total funds ) ​ ​

Step 2: Add all debits (expenses): 89.75 ( electric bill ) + 64.38 ( groceries ) 154.13 ( total debits ) \begin{array}{rl}

  • 89.75 & (\text{electric bill}) \\
  • + 64.38 & (\text{groceries}) \\
  • \hline
  • \mathbf{154.13} & (\text{total debits})
  • \end{array} 89.75 + 64.38 154.13 ​ ( electric bill ) ( groceries ) ( total debits ) ​ ​
  • Step 3: Subtract total debits from total funds: 691.75 − 154.13 537.62 \begin{array}{r}
  • 691.75 \\
  • 154.13 \\
  • \hline
  • \mathbf{537.62}
  • \end{array} 691.75 − 154.13 537.62 ​ ​

Final Answer: Alex's ending account balance is $537.62 .

ACCUPLACER Exam Traps & Common Errors

Warning

Trap 1: The 'Subtracting Down' Borrowing Mistake When subtracting 8.4 − 2.67 8.4 - 2.67 8.4 − 2.67 , students who fail to annex zeros write 8.40 − 2.67 8.40 - 2.67 8.40 − 2.67 incorrectly by subtracting 7 − 0 = 7 7 - 0 = 7 7 − 0 = 7 in the hundredths column, getting 5.77 5.77 5.77 instead of borrowing ( 10 − 7 = 3 10 - 7 = 3 10 − 7 = 3 ) to get 5.73 5.73 5.73 .

Warning

Trap 2: Misaligning Whole Numbers and Decimals When adding a whole number to a decimal (e.g., 15 + 3.45 15 + 3.45 15 + 3.45 ), remember that whole numbers have an unwritten decimal point at their far right ( 15.00 15.00 15.00 ). Adding 15 + 3.45 15 + 3.45 15 + 3.45 as 0.15 + 3.45 0.15 + 3.45 0.15 + 3.45 yields a catastrophic error.